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Calculus

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Calculus. Sterling Lehmitz Kenzie Rhodesie Brennan Perry. Integration and Finding Areas Under Curves. What is the Key to Finding the Area Under Curves?. Parts of the Integral. What Does This Area Mean?.

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  1. Calculus Sterling Lehmitz Kenzie Rhodesie Brennan Perry Integration and Finding Areas Under Curves.

  2. What is the Key to Finding the Area Under Curves?

  3. Parts of the Integral

  4. What Does This Area Mean? • If the function represents velocity, the area under the curve is the displacement of the object. • It can be used in finding the volume of objects defined by curves. • It also means that Sterling should one day rule the galaxy. • Sort of.

  5. How Do Integrals Work? ANTIDERIVATIVES What is an antiderivative? then antiderivatives are going backwards in these series (not counting initial position).

  6. What Do You Do With Antiderivatives? • You evaluate the integral using a and b. • If the antiderivative of f(x) is F(x), then you evaluate the integral like so:

  7. An Example

  8. This is also equal to the area under the graph from 0 to 5.

  9. Good to Remember

  10. You can also find the area between two different curves. • Start by finding where they intersect with each other. • The intersection points (x-values [or y-values if the function is inverted]) are the limits of integration. • Then, within the integral, you take the first (top [or right]) function and subtract the second (bottom [or left]) function. • Then you solve the integral and evaluate it from the limits you found from the intersection points.

  11. Another example • Find the area bounded by these functions:

  12. Solving It • Set the functions equal to each other to find where they intersect. • So the functions intersect at x = -2 and x = 1.

  13. Tricky Integration • Suppose you run into something that you can’t easily find the antiderivative of. • Well, there are some different ways to solve these integrals. TABULAR INTEGRATION AND INTEGRATION BY PARTS

  14. The Tabular Method • If there is a power function (can be differentiated repeatedly until it becomes 0) and something that can easily be integrated repeatedly, then we can do tabular integration. • This is integrating using a table. That’s right, YOU CAN’T!

  15. So We Use a Table to Integrate Like so:

  16. The Other Method for Integrating Difficult Integrals is Integration by Parts • Let’s use the same function from tabular integration to explain this. • Thinking of the product rule and then integrating it with respect to x, we can get

  17. An Example of Integration by Parts

  18. Another Example of Integration by Parts • Here is where everything goes during the integration.

  19. If You Want to Know More… • http://www.maths.abdn.ac.uk/~igc/tch/ma1002/int/int.html